Found problems: 2265
A closed pentagonal line is inscribed in a sphere of the diameter $1$, and has all edges of length $\ell$.
Prove that $\ell \le \sin \frac{2\pi}{5}$
.
What is the maximum possible volume of a cylinder inscribed in a cone and having the same axis of symmetry as the cone? What is the maximum possible volume of a ball inscribed in the cone with center on the axis of symmetry of the cone? Compare these three volumes.
Let $p$ be a fixed prime. Determine all the integers $m$, as function of $p$, such that there exist $a_1, a_2, \ldots, a_p \in \mathbb{Z}$ satisfying
\[m \mid a_1^p + a_2^p + \cdots + a_p^p - (p+1).\]
In the plane there is a worm of length 1. Prove that it can be always covered by means of half of a circular disk of diameter 1.
[i]Note.[/i] Under a "worm", we understand a continuous curve. The "half of a circular disk" is a semicircle including its boundary.
Akash's birthday cake is in the form of a $4 \times 4 \times 4$ inch cube. The cake has icing on the top and the four side faces, and no icing on the bottom. Suppose the cake is cut into $64$ smaller cubes, each measuring $1 \times 1 \times 1$ inch, as shown below. How many of the small pieces will have icing on exactly two sides?
[asy]
/*
Created by SirCalcsALot and sonone
Code modfied from https://artofproblemsolving.com/community/c3114h2152994_the_old__aops_logo_with_asymptote
*/
import three;
currentprojection=orthographic(1.75,7,2);
//++++ edit colors, names are self-explainatory ++++
//pen top=rgb(27/255, 135/255, 212/255);
//pen right=rgb(254/255,245/255,182/255);
//pen left=rgb(153/255,200/255,99/255);
pen top = rgb(170/255, 170/255, 170/255);
pen left = rgb(81/255, 81/255, 81/255);
pen right = rgb(165/255, 165/255, 165/255);
pen edges=black;
int max_side = 4;
//+++++++++++++++++++++++++++++++++++++++
path3 leftface=(1,0,0)--(1,1,0)--(1,1,1)--(1,0,1)--cycle;
path3 rightface=(0,1,0)--(1,1,0)--(1,1,1)--(0,1,1)--cycle;
path3 topface=(0,0,1)--(1,0,1)--(1,1,1)--(0,1,1)--cycle;
for(int i=0; i<max_side; ++i){
for(int j=0; j<max_side; ++j){
draw(shift(i,j,-1)*surface(topface),top);
draw(shift(i,j,-1)*topface,edges);
draw(shift(i,-1,j)*surface(rightface),right);
draw(shift(i,-1,j)*rightface,edges);
draw(shift(-1,j,i)*surface(leftface),left);
draw(shift(-1,j,i)*leftface,edges);
}
}
picture CUBE;
draw(CUBE,surface(leftface),left,nolight);
draw(CUBE,surface(rightface),right,nolight);
draw(CUBE,surface(topface),top,nolight);
draw(CUBE,topface,edges);
draw(CUBE,leftface,edges);
draw(CUBE,rightface,edges);
// begin made by SirCalcsALot
int[][] heights = {{4,4,4,4},{4,4,4,4},{4,4,4,4},{4,4,4,4}};
for (int i = 0; i < max_side; ++i) {
for (int j = 0; j < max_side; ++j) {
for (int k = 0; k < min(heights[i][j], max_side); ++k) {
add(shift(i,j,k)*CUBE);
}
}
}
[/asy]
$\textbf{(A)}\ 12\qquad~~\textbf{(B)}\ 16\qquad~~\textbf{(C)}\ 18\qquad~~\textbf{(D)}\ 20\qquad~~\textbf{(E)}\ 24$
Let $P$ be a polyhedron whose faces are colored black and white so that there are more black faces and no two black faces are adjacent. Show that $P$ is not circumscribed about a sphere.
Let $ABCD$ be a regular tetrahedron with side $a$. Take $E,E'$ on the edge $AB, F, F'$ on the edge $AC$ and $G,G'$ on the edge AD so that $AE =a/6,AE' = 5a/6,AF= a/4,AF'= 3a/4,AG = a/3,AG'= 2a/3$. Compute the volume of $EFGE'F'G'$ in term of $a$ and find the angles between the lines $AB,AC,AD$ and the plane $EFG$.
This year's gift idea from BabyMath consists of a series of nine colored plastic containers of decreasing size, alternating in shape like a cube and a sphere. All containers can open and close with a convenient hinge, and each container can hold just about anything next in line. The largest and smallest container are both cubes. Determine the relationship between the edge lengths of these cubes.
Prove that if a regular icosahedron and a regular dodecahedron have a common circumsphere, then they have a common insphere.
[i](7 points)[/i]
In a country, towns are connected by roads. Each town is directly connected to exactly three other towns. Show that there exists a town from which you can make a round-trip, without using the same road more than once, and for which the number of roads used is not divisible by $3$. (Not all towns need to be visited.)
The number of ordered pairs of integers $ (m,n)$ for which $ mn \ge 0$ and \[m^3 \plus{} n^3 \plus{} 99mn \equal{} 33^3\] is equal to
$ \textbf{(A)}\ 2\qquad
\textbf{(B)}\ 3\qquad
\textbf{(C)}\ 33\qquad
\textbf{(D)}\ 35\qquad
\textbf{(E)}\ 99$
In tetrahedron $ABCD$, let $V$ be the volume of the tetrahedron and $R$ the radius of the sphere that it tangent to all four faces of the tetrahedron. Let $P$ be the surface area of the tetrahedron. Prove that $$r=\frac{3V}{P}.$$
[i]Proposed by CaptainFlint.[/i]
Two perpendicular planes intersect a sphere in two circles. These circles intersect in two points, $A$ and $B$, such that $AB=42$. If the radii of the two circles are $54$ and $66$, find $R^2$, where $R$ is the radius of the sphere.
A box whose shape is a parallelepiped can be completely filled with cubes of side $1.$ If we put in it the maximum possible number of cubes, each of volume $2$, with the sides parallel to those of the box, then exactly $40$ percent of the volume of the box is occupied. Determine the possible dimensions of the box.
In a convex $n$-gonal prism all sides are equal. For what $n$ is this prism right?
Suppose that the sum of the squares of two complex numbers $x$ and $y$ is 7 and the sum of the cubes is 10. What is the largest real value that $x + y$ can have?
Let $u_{jk}$ be a real number for each $j=1,2,3$ and each $k=1,2$ and let $N$ be an integer such that
\[\max_{1\le k \le 2} \sum_{j=1}^3 |u_{jk}| \leq N\]
Let $M$ and $l$ be positive integers such that $l^2 <(M+1)^3$. Prove that there exist integers $\xi_1,\xi_2,\xi_3$ not all zero, such that
\[\max_{1\le j \le 3}\xi_j \le M\ \ \ \ \text{and} \ \ \ \left|\sum_{j=1}^3 u_{jk}\xi_k\right| \le \frac{MN}{l} \ \ \ \ \text{for k=1,2}\]
Corners are sliced off a unit cube so that the six faces each become regular octagons. What is the total volume of the removed tetrahedra?
$ \textbf{(A)}\ \frac {5\sqrt {2} \minus{} 7}{3}\qquad \textbf{(B)}\ \frac {10 \minus{} 7\sqrt {2}}{3}\qquad \textbf{(C)}\ \frac {3 \minus{} 2\sqrt {2}}{3}\qquad \textbf{(D)}\ \frac {8\sqrt {2} \minus{} 11}{3}\qquad \textbf{(E)}\ \frac {6 \minus{} 4\sqrt {2}}{3}$
A cube has eight vertices (corners) and twelve edges. A segment, such as $x$, which joins two vertices not joined by an edge is called a diagonal. Segment $y$ is also a diagonal. How many diagonals does a cube have?
[asy]draw((0,3)--(0,0)--(3,0)--(5.5,1)--(5.5,4)--(3,3)--(0,3)--(2.5,4)--(5.5,4));
draw((3,0)--(3,3));
draw((0,0)--(2.5,1)--(5.5,1)--(0,3)--(5.5,4),dashed);
draw((2.5,4)--(2.5,1),dashed);
label("$x$",(2.75,3.5),NNE);
label("$y$",(4.125,1.5),NNE);
[/asy]
$\textbf{(A)}\ 6 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ 14 \qquad \textbf{(E)}\ 16$
Sphere $S$ is inscribed in cone $C$. The height of $C$ equals its radius, and both equal $12+12\sqrt2$. Let the vertex of the cone be $A$ and the center of the sphere be $B$. Plane $P$ is tangent to $S$ and intersects $\overline{AB}$. $X$ is the point on the intersection of $P$ and $C$ closest to $A$. Given that $AX=6$, find the area of the region of $P$ enclosed by the intersection of $C$ and $P$.
The altitudes $AA_1,BB_1,CC_1,DD_1$ of a tetrahedron $ABCD$ intersect in the center $H$ of the sphere inscribed in the tetrahedron $A_1B_1C_1D_1$. Prove that the tetrahedron $ABCD$ is regular.
(D. Tereshin)
There are $ n$ points ($ n \geq 4$) on a sphere with radius $ R$, and not all of them lie on the same semi-sphere. Prove that among all the angles formed by any two of the $ n$ points and the sphere centre $ O$ ($ O$ is the vertex of the angle), there is at least one that is not less than $ \displaystyle 2 \arcsin{\frac{\sqrt{6}}{3}}$.
Prove that the number $1\underbrace{\hbox{0...0}}_{\hbox{49}}5\underbrace{\hbox{0...0}}_{\hbox{99}}1$ is not the cube of any integer.
Given a cube. On each edge of the cube, we write a number, either $1$ or $-1$. For each face of the cube, we multiply the four numbers on the edges of this face, and write the product on this face. Finally, we add all the eighteen numbers that we wrote down on the edges and face of the cube.
What is the smallest possible sum that we can get?
Prove that the volume $V$ and the lateral area $S$ of a right circular cone satisfy the inequality
\[\left( \frac{6V}{\pi}\right)^2 \leq \left( \frac{2S}{\pi \sqrt 3}\right)^3\]
When does equality occur?